Solving forward-backward stochastic differential equations explicitly — a four step scheme

Abstract

In this paper we i n v estigate the nature of the adapted solutions to a class of forward-backward stochastic dierential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an \ordinary" sense over an arbitrarily prescribed time duration, via a direct \Four Step Scheme". Using this scheme, we further prove that the backward components of the adapted solution are determined explicitly by the forward component via the solution of a certain quasi-linear parabolic PDE system. Moreover the uniqueness of the adapted solutions (over an arbitrary time duration), as well as the continuous dependence of the solutions on the parameters, can all be proved within this unied framework. Some special cases are studied separately. In particular, we derive a new form of the integral representation of the Clark-Haussmann-Ocone t ype for functionals (or functions) of diusions, in which the conditional expectation is no longer needed.

Singular FBSDEs and scalar conservation laws driven by diffusion processes

“…Fundamental existence and uniqueness results for fully-coupled FBSDEs can be found in the subsequent works of Ma, Protter and Yong [12], Peng and Wu [16] and Delarue [6], or in the book of Ma and Yong [13]. All these results remain rather technical in nature.…”

Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients

“…In view of Ma, Protter and Yong's work [4], we can try to use another type of FBSDE to represent solutions of equation (5). Indeed, Ma, Protter and Yong studied the relationship between solutions of the FBSDE + f t, x, u(t, x) , ∇u(t, x) + g t, x, u(t, x) = 0 for t ∈ (0, T ),…”

On the homotopy analysis method for backward/forward-backward stochastic differential equations

“…In 1994, Ma, Protter and Yong [15] proposed a four step scheme to numerically solve the corresponding parabolic partial differential equations of FBSDEs. Based on the four step scheme, some numerical algorithms [16][17][18][19] are proposed.…”